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Preservice teachers commonly struggle with topics related to fractions, ratio, and proportion. The use of double number lines can facilitate the teaching in these areas and provide autonomy and flexibility in mathematical thinking. Moreover, double number lines support many topics in the curriculum, including unit rate, convenience scaling, and least common multiple.

The domain of Mathematical Knowledge for Teaching (MKT) has provided theoretical and empirical advancements to teacher education scholarship. Today, MKT is widely recognized as a critical component of teachers' preparation, professional development, and experience. Early studies at the elementary level set the stage for defining, measuring, assessing, and augmenting this knowledge in pre-service and in-service teachers. Subsequent studies are demonstrating MKT's significance in university and secondary teaching. Despite MKT's growing influence in education, one institution that has been largely ignored is the community college. Given the community college's role of remediation alongside the diverse and often underprepared student body it serves, we urge the mathematics education community to examine MKT in the context of community college developmental mathematics. Based on preliminary interviews with experienced faculty at two-year institutions (n = 12), we propose a provisional amendment to the domains of MKT -- the Caring Map. We speculate that in order to make sense of developmental mathematics teaching at the community college, research is needed to determine the extent to which teacher caring supports mathematics teaching, student learning, and student achievement.

The Corvette Problem is a nonroutine investigation, one that provides an authentic context to explore many interrelated calculus ideas. In our years of sharing this problem with students and colleagues, we have found additional topics from the calculus curriculum to have rich interpretations in this environment. The Corvette Problem contextualizes much of the material in first-semester undergraduate calculus by way of unifying theorems and definitions into a powerful mosaic of ideas.

What students believe about the nature and role of mathematics is critical to their learning of mathematics, as negative beliefs stunt children's curiosity about mathematics and cause student motivations to learn mathematics to be based on grades and other extrinsic motivators (Howley et al., 2005). For meaningful learning to take place, students must have access to a challenging, process-based curriculum that is differentiated for their learning needs and exposes them to the authentic behaviors of a mathematician so that they can form deeper understandings of mathematics concepts and the nature of mathematics. At this point in time, no controlled studies have been completed that explore the effects of the use of self-paced, challenging, process-based curricula on talented students' beliefs about mathematics. This research seeks to establish whether there is a relationship between students' beliefs about mathematics and their images of a self-paced, challenging, process-based curriculum. A nonexperimental, survey research design was employed. The respondents are 39 gifted sixth grade students from a large suburban district in Southeast Florida which had adopted the "Elements of Mathematics: Foundations" ("EMF") curriculum for its most talented middle school mathematics students. Overall, these talented students expressed positive views of mathematics. They strongly agreed that mathematics is thought-provoking, makes a unique contribution to human knowledge, connected to the real world, and supports many different ways of looking at and solving problems. The students found "EMF" to be gratifying and challenging. Most reported that "EMF" better prepared them for high school and college mathematics courses than an accelerated, traditional curriculum and that "EMF" improved their self-study skills, ability to analyze complex problems and focus for extended periods of time, and confidence with respect to challenging programs.

This article highlights three of the eight Mathematics Teaching Practices (MTP) published in the National Council of Teachers of Mathematics' (NCTM's) "Principles to Actions: Ensuring Mathematical Success for All" (2014): (1) facilitating meaningful mathematical discourse (MTP 4); (2) posing purposeful questions (MTP 5); and (3) eliciting and using evidence of student thinking (MTP 8). In this article, the authors discuss the meaning of the term "active learning" and discuss the five practices (Smith and Stein 2011) as a particularly illuminating model. The five practices offer a powerful framework that the authors have used to activate their mathematics classrooms. They share two vignettes of classroom learning from a first-year calculus class at a small university. Such teaching and learning experiences span all levels, K-16. The article's focus is on the implementation and management of active instructional practices, irrespective of mathematical content. The article includes examples of student feedback from having experienced active learning in their college mathematics class.

Sometimes the best mathematics problems come from the most unexpected situations. Last summer, a Corvette raced down a local quarter-mile drag strip. The driver, a family member, provided the spectators with time and distance-traveled data from his time slip and asked "Can you calculate how many seconds it took me to go from 0 to 60 mph?" Although initially this question seemed like a straightforward one, it was soon clear that depending on the solution strategy and assumptions, different answers were possible. Thus began the ongoing discussions with colleagues--and with high school mathematics teacher friends over pizza and with mechanical engineer family members at holiday dinners--to collectively decide on the "best" method. The mathematical discussions that arose on how to best solve the problem prompted two questions: (1) What makes this problem so intriguing? and (2) What would students do? Any interesting mathematical task will likely encourage teachers to wonder what aspects of the task make it special. The authors of this article wanted to know why this problem generated these mathematical conversations and how they could incorporate it into a calculus class. Insights from colleagues and students revealed several qualities of the problem that they believe contribute to its intrigue and worth. What they found illustrates what they consider to be three hallmarks of a good problem: (1) The problem solver must decide what mathematics to introduce; (2) The task users real-life data; and (3) The task requires mathematical modeling. A nonroutine task with three hallmarks of a good problem offers the flexibility to model real-life, messy data.

In this article on introductory calculus, intriguing questions are generated that can ignite an appreciation for the subject of mathematics. These questions open doors to advanced mathematical thinking and harness many elements of research-oriented mathematics. Such questions also offer greater incentives for students to think and reflect. Research has shown that nonroutine problems in algebra may alter students' impressions of mathematics and allow for a keener sense of understanding and more productive lines of thinking. Perhaps the greatest benefit to teachers is seeing the application of important mathematical ideas outside the narrow context in which they are often introduced. An open-ended calculus problem is presented that spurs novelties in student thinking and shows the benefits of timely teacher intervention.

A central concern in science education is promoting inquiry activities, and a significant amount of research in science education has addressed what inhibits this goal. However, this research has tended to be focused on implementation issues. Levels of implementation are correlated with extrinsic barriers such as time, resources and teacher knowledge, but the internal structure of inquiry remains unexamined. However, are there also some ways in which inquiry based instruction is intrinsically difficult? That is, given perfect conditions for implementation, is devising inquiry activities a non-trivial matter, and are there patterns to the difficulties in doing so? The aim of this paper is to provide both a theoretical and empirical basis for the intrinsic problems associated with the construction of inquiry activities. We present a theoretical framework that connects contemporary studies of science to the classroom context. We then examine a case study of teacher education students designing inquiry-based instruction to demonstrate empirical evidence of intrinsic problems. Our ultimate goal is to improve our ability as teacher educators to guide science teachers in fulfilling this central task of their work.

As a college mathematics teacher for six years, the author has learned his share of pedagogical techniques through research, practice, and reflection. He adopts new theories into classroom practice, differentiates instruction when necessary, and remains open to the diversity his students bring with respect to academic readiness, cultural norms, and learning styles. Through these experiences, the author has learned what typically works and what needs attention. But he learned his biggest lesson just recently with his calculus 2 class. While approaching the end of a unit on infinite series, he did his usual brainstorming for an optional assignment--something to pique student interest in this unit. In this article, the author first discusses two student responses to share the fruits of their efforts. Second, he includes some anecdotes about what his students gained from this experience. The author closes with some of his realizations from having done this activity. (Contains 2 figures.)

Preservice teachers commonly struggle with topics related to fractions, ratio, and proportion. The use of double number lines can facilitate the teaching in these areas and provide autonomy and flexibility in mathematical thinking. Moreover, double number lines support many topics in the curriculum, including unit rate, convenience scaling, and least common multiple. [ABSTRACT FROM AUTHOR]

Talented middle school students in most public schools and traditional private schools do not have access to a curriculum that allows them to go at their own pace, adequately challenges them, or presents mathematics in a process-based way so that students can become familiar with behaviors of mathematicians (Gentry, Gable, & Springer, 2000; Rogers, 2007). The age of the student was statistically significantly related to level of agreement with the statements "Mathematics supports many different ways of looking at and solving the same problems" (t(35) = 2.94, p = .01), "Math is intricately connected to the real world" (t(35) = 3.81, p = .00), and "In mathematics you can be creative" (t(35) = 5.85, p = .00). About 84% of those students who said one of their favorite subjects was mathematics selected "doing a puzzle" on the item "Ideally, mathematics is like" while only roughly 29% of those students who did not say that mathematics was one of their favorite subjects picked "doing a puzzle". Overall, the students who found EMF most enjoyable and rewarding, reported that EMF had improved their confidence, further developed their ability to focus and their self-study skills, and had made them more ready for future mathematics courses had more positive views of mathematics. [Extracted from the article]

The article presents answers to various questions related to calculus including the association between cancellation of factors and removable discontinuity, the use of midpoint formula in finding infection point, and the meaning of numbers in function formula.

The article offers the author's insights regarding the aspects of mathematical solutions through integration considering various techniques including substitution, partial fractions, and integration by parts.

The author discusses the article titled "Is Algebra Necessary?," by Andrew Hacker of the journal "New York Times." Topics include the criticality of variable concept in mathematical endeavors, the role of algebra in technological developments such as medical imaging and digital media, and the debates and arguments among people who agree and who does not agree on the necessity of teaching algebra.

The article presents information on a report which suggests that nonroutine problems in algebra may alter impressions of mathematics for students and allows a keener sense of understanding and more productive lines of thinking. It suggests mathematics teachers to involve nonroutine calculus task in the present curricula that will help in developing students' approach of dealing with a mathematical problem.

The article proposes a four-strand research agenda offering a vision for research in community college mathematics education that connects to American Mathematical Association of Two-Year Colleges' (AMATYC) strategic plan. The ways in which mathematics education may inform the work of community college mathematics teachers is discussed. It also discusses the need for a research agenda while also promoting a call to action for impacting mathematics teaching and learning.

The article discusses a classic related rates problem along with the solution found in undergraduate calculus textbooks. The problem presented involves the rate of increase in the distance between the balloon and its owner. The standard solution consists first of sketching a picture of the situation, writing down the information that is explicitly provided and what needs to be found. The solution created by a student expresses the connection between the variables, finds an ordinary derivative unrelated to time and multiplies by the other moving part.

The article focuses on the use of the Mean Value Theorems from integrated and differentiated calculus in finding the average value of a continuous function on an interval. A new equation is proposed by the author based on the notion that the area under the curve of can be represented as F(b) - F(a) in y + ƒ(x) from a to b. The alternative solution exposes the students to properties of integration and Fundamental Theorem of Calculus.